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2: Symbols for the Elements

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13.6 Dalton’s Law of Partial Pressures

427

SOLUTION

Where Are We Going?

We want to determine the partial pressure of helium and oxygen and the

total pressure in the tank.

What Do We Know?

• We know the initial volume, pressure, and temperature of both

gases.

• We know the final volume of the tank.

• The temperature remains constant.

• Ideal gas law: PV ϭ nRT.

• Dalton’s law of partial pressures: Ptotal ϭ P1 ϩ P2 ϩ p

What Information Do We Need?

• R ϭ 0.08206 L atm/mol K.

How Do We Get There?

MATH SKILL BUILDER

PV ϭ nRT

PV

nRT

ϭ

RT

RT

PV

ϭn

RT

Because the partial pressure of each gas depends on the moles of that gas present, we must first calculate the number of moles of each gas by using the

ideal gas law in the form

PV

RT

From the above description we know that P ϭ 1.0 atm, V ϭ 12 L for O2 and

46 L for He, and T ϭ 25 ϩ 273 ϭ 298 K. Also, R ϭ 0.08206 L atm/K mol (as

always).

Moles of O2 ϭ nO2 ϭ

Moles of He ϭ nHe ϭ

(1.0 atm)(12 L)

(0.08206 L atmրK mol)(298 K)

(1.0 atm)(46 L)

(0.08206 L atmրK mol)(298 K)

ϭ 0.49 mol

ϭ 1.9 mol

The tank containing the mixture has a volume of 5.0 L, and the temperature

is 25 °C (298 K). We can use these data and the ideal gas law to calculate the

partial pressure of each gas.

PO2 ϭ

PHe ϭ

nRT

V

(0.49 mol)(0.08206 L atmրK mol)(298 K)

5.0 L

(1.9 mol)(0.08206 L atmրK mol)(298 K)

5.0 L

ϭ 2.4 atm

ϭ 9.3 atm

The total pressure is the sum of the partial pressures.

Kurt Amsler

Ptotal ϭ PO2 ϩ PHe ϭ 2.4 atm ϩ 9.3 atm ϭ 11.7 atm

Divers use a mixture of oxygen

and helium in their breathing

tanks when diving to depths

greater than 150 feet.

R E A L I T Y C H E C K The volume of each gas decreased, and the pressure of

each gas increased. The partial pressure of helium is greater than that of oxygen, which makes sense because the initial temperatures and pressures of helium and oxygen were the same, but the initial volume of helium was much

greater than that of oxygen.

428 Chapter 13 Gases

Oxygen plus

water vapor

KClO3

Figure 13.12

The production of oxygen by

thermal decomposition of KClO3.

Self-Check EXERCISE 13.9 A 2.0-L flask contains a mixture of nitrogen gas and oxygen gas at 25 °C. The

total pressure of the gaseous mixture is 0.91 atm, and the mixture is known

to contain 0.050 mol N2. Calculate the partial pressure of oxygen and the

moles of oxygen present.

See Problems 13.67 through 13.70. ■

Table 13.2 The Vapor

Pressure of Water as a

Function of Temperature

T (°C)

P (torr)

0.0

4.579

10.0

9.209

20.0

17.535

25.0

23.756

30.0

31.824

40.0

55.324

60.0

149.4

70.0

233.7

90.0

525.8

EXAMPLE 13.13

A mixture of gases occurs whenever a gas is collected by displacement

of water. For example, Figure 13.12 shows the collection of the oxygen gas

that is produced by the decomposition of solid potassium chlorate. The gas

is collected by bubbling it into a bottle that is initially filled with water. Thus

the gas in the bottle is really a mixture of water vapor and oxygen. (Water vapor is present because molecules of water escape from the surface of the liquid and collect as a gas in the space above the liquid.) Therefore, the total

pressure exerted by this mixture is the sum of the partial pressure of the gas

being collected and the partial pressure of the water vapor. The partial pressure of the water vapor is called the vapor pressure of water. Because water

molecules are more likely to escape from hot water than from cold water, the

vapor pressure of water increases with temperature. This is shown by the values of vapor pressure at various temperatures in Table 13.2.

Using Dalton’s Law of Partial Pressures, II

A sample of solid potassium chlorate, KClO3, was heated in a test tube (see

Figure 13.12) and decomposed according to the reaction

2KClO3(s) S 2KCl(s) ϩ 3O2(g)

The oxygen produced was collected by displacement of water at 22 °C. The resulting mixture of O2 and H2O vapor had a total pressure of 754 torr and a volume of 0.650 L. Calculate the partial pressure of O2 in the gas collected and the

number of moles of O2 present. The vapor pressure of water at 22 °C is 21 torr.

SOLUTION

Where Are We Going?

We want to determine the partial pressure of oxygen collected by water displacement and the number of moles of O2 present.

What Do We Know?

• We know the temperature, total pressure, and volume of gas

collected by water displacement.

13.7 Laws and Models: A Review

429

• We know the vapor pressure of water at this temperature.

• Ideal gas law: PV ϭ nRT.

• Dalton’s law of partial pressures: Ptotal ϭ P1 ϩ P2 ϩ p

What Information Do We Need?

• R ϭ 0.08206 L atm/mol K.

How Do We Get There?

We know the total pressure (754 torr) and the partial pressure of water

(vapor pressure ϭ 21 torr). We can find the partial pressure of O2 from Dalton’s law of partial pressures:

Ptotal ϭ PO2 ϩ PH2O ϭ PO2 ϩ 21 torr ϭ 754 torr

or

PO2 ϩ 21 torr ϭ 754 torr

We can solve for PO2 by subtracting 21 torr from both sides of the equation.

PO2 ϭ 754 torr Ϫ 21 torr ϭ 733 torr

Next we solve the ideal gas law for the number of moles of O2.

nO2 ϭ

PO2V

RT

In this case, PO2 ϭ 733 torr. We change the pressure to atmospheres as follows:

MATH SKILL BUILDER

PV ϭ nRT

PV

nRT

ϭ

RT

RT

PV

ϭn

RT

733 torr

ϭ 0.964 atm

760 torr/atm

Then,

V ϭ 0.650 L

T ϭ 22 °C ϭ 22 ϩ 273 ϭ 295 K

R ϭ 0.08206 L atm/K mol

so

nO2 ϭ

(0.964 atm)(0.650 L)

(0.08206 L atmրK mol)(295 K)

ϭ 2.59 ϫ 10Ϫ2 mol

Self-Check EXERCISE 13.10 Consider a sample of hydrogen gas collected over water at 25 °C where the

vapor pressure of water is 24 torr. The volume occupied by the gaseous mixture is 0.500 L, and the total pressure is 0.950 atm. Calculate the partial pressure of H2 and the number of moles of H2 present.

See Problems 13.71 through 13.74. ■

13.7 Laws and Models: A Review

OBJECTIVE:

To understand the relationship between laws and models (theories).

In this chapter we have considered several properties of gases and have seen

how the relationships among these properties can be expressed by various

laws written in the form of mathematical equations. The most useful of these

is the ideal gas equation, which relates all the important gas properties. However, under certain conditions gases do not obey the ideal gas equation. For

430 Chapter 13 Gases

example, at high pressures and/or low temperatures, the properties of gases

deviate significantly from the predictions of the ideal gas equation. On the

other hand, as the pressure is lowered and/or the temperature is increased,

almost all gases show close agreement with the ideal gas equation. This

means that an ideal gas is really a hypothetical substance. At low pressures

and/or high temperatures, real gases approach the behavior expected for an

ideal gas.

At this point we want to build a model (a theory) to explain why a gas

behaves as it does. We want to answer the question, What are the characteristics of the individual gas particles that cause a gas to behave as it does? However,

before we do this let’s briefly review the scientific method. Recall that a law

is a generalization about behavior that has been observed in many experiments. Laws are very useful; they allow us to predict the behavior of similar

systems. For example, a chemist who prepares a new gaseous compound can

assume that that substance will obey the ideal gas equation (at least at low P

and/or high T ).

However, laws do not tell us why nature behaves the way it does. Scientists try to answer this question by constructing theories (building models). The models in chemistry are speculations about how individual atoms

or molecules (microscopic particles) cause the behavior of macroscopic systems (collections of atoms and molecules in large enough numbers so that

we can observe them).

A model is considered successful if it explains known behavior and predicts correctly the results of future experiments. But a model can never be

proved absolutely true. In fact, by its very nature any model is an approximation and is destined to be modified, at least in part. Models range from the

simple (to predict approximate behavior) to the extraordinarily complex (to

account precisely for observed behavior). In this text, we use relatively simple models that fit most experimental results.

13.8 The Kinetic Molecular Theory of Gases

OBJECTIVE:

Module 16: Gas Law and

the Kinetic Molecular Theory covers

concepts in this section.

To understand the basic postulates of the kinetic molecular theory.

A relatively simple model that attempts to explain the behavior of an ideal

gas is the kinetic molecular theory. This model is based on speculations

about the behavior of the individual particles (atoms or molecules) in a gas.

The assumptions (postulates) of the kinetic molecular theory can be stated

as follows:

Postulates of the Kinetic Molecular Theory of Gases

1. Gases consist of tiny particles (atoms or molecules).

2. These particles are so small, compared with the distances between them,

that the volume (size) of the individual particles can be assumed to be

negligible (zero).

3. The particles are in constant random motion, colliding with the walls of the

container. These collisions with the walls cause the pressure exerted by the

gas.

4. The particles are assumed not to attract or to repel each other.

5. The average kinetic energy of the gas particles is directly proportional to

the Kelvin temperature of the gas.

13.9 The Implications of the Kinetic Molecular Theory

431

The kinetic energy referred to in postulate 5 is the energy associated

with the motion of a particle. Kinetic energy (KE) is given by the equation

KE ϭ 21 mv2, where m is the mass of the particle and v is the velocity (speed)

of the particle. The greater the mass or velocity of a particle, the greater its

kinetic energy. Postulate 5 means that if a gas is heated to higher temperatures, the average speed of the particles increases; therefore, their kinetic energy increases.

Although real gases do not conform exactly to the five assumptions

listed here, we will see in the next section that these postulates do indeed explain ideal gas behavior—behavior shown by real gases at high temperatures

and/or low pressures.

13.9 The Implications of the Kinetic

Molecular Theory

OBJECTIVES:

To understand the term temperature. • To learn how the kinetic molecular theory explains the gas laws.

In this section we will discuss the qualitative relationships between the kinetic molecular (KM) theory and the properties of gases. That is, without going into the mathematical details, we will show how the kinetic molecular

theory explains some of the observed properties of gases.

The Meaning of Temperature

In Chapter 2 we introduced temperature very practically as something we

measure with a thermometer. We know that as the temperature of an object

increases, the object feels “hotter” to the touch. But what does temperature

really mean? How does matter change when it gets “hotter”? In Chapter 10

we introduced the idea that temperature is an index of molecular motion.

The kinetic molecular theory allows us to further develop this concept. As

postulate 5 of the KM theory states, the temperature of a gas reflects how

rapidly, on average, its individual gas particles are moving. At high temperatures the particles move very fast and hit the walls of the container frequently, whereas at low temperatures the particles’ motions are more sluggish and they collide with the walls of the container much less often.

Therefore, temperature really is a measure of the motions of the gas particles.

In fact, the Kelvin temperature of a gas is directly proportional to the average kinetic energy of the gas particles.

The Relationship Between Pressure and Temperature

To see how the meaning of temperature given above helps to explain gas behavior, picture a gas in a rigid container. As the gas is heated to a higher temperature, the particles move faster, hitting the walls more often. And, of

course, the impacts become more forceful as the particles move faster. If the

pressure is due to collisions with the walls, the gas pressure should increase

as temperature is increased.

Is this what we observe when we measure the pressure of a gas as it is

heated? Yes. A given sample of gas in a rigid container (if the volume is not

changed) shows an increase in pressure as its temperature is increased.

432 Chapter 13 Gases

Pext

Pext

Increase in

temperature

a

A gas confined in a

cylinder with a movable

piston. The gas pressure

Pgas is just balanced by

the external pressure Pext.

That is, Pgas ϭ Pext.

Figure 13.13

b

The temperature of the

gas is increased at

constant pressure Pext.

The increased particle

motions at the higher

temperature push back

the piston, increasing the

volume of the gas.

The Relationship Between Volume and Temperature

Now picture the gas in a container with a movable piston. As shown in Figure 13.13a, the gas pressure Pgas is just balanced by an external pressure Pext.

What happens when we heat the gas to a higher temperature? As the temperature increases, the particles move faster, causing the gas pressure to increase. As soon as the gas pressure Pgas becomes greater than Pext (the pressure

holding the piston), the piston moves up until Pgas ϭ Pext. Therefore, the KM

model predicts that the volume of the gas will increase as we raise its temperature at a constant pressure (Figure 13.13b). This agrees with experimental observations (as summarized by Charles’s law).

EXAMPLE 13.14

Using the Kinetic Molecular Theory

to Explain Gas Law Observations

Use the KM theory to predict what will happen to the pressure of a gas when

its volume is decreased (n and T constant). Does this prediction agree with

the experimental observations?

SOLUTION

When we decrease the gas’s volume (make the container smaller), the particles hit the walls more often because they do not have to travel so far between the walls. This would suggest an increase in pressure. This prediction

on the basis of the model is in agreement with experimental observations of

gas behavior (as summarized by Boyle’s law). ■

In this section we have seen that the predictions of the kinetic molecular theory generally fit the behavior observed for gases. This makes it a useful and successful model.

13.10

OBJECTIVES:

Gas Stoichiometry

To understand the molar volume of an ideal gas. • To learn the definition

of STP. • To use these concepts and the ideal gas equation.

We have seen repeatedly in this chapter just how useful the ideal gas equation is. For example, if we know the pressure, volume, and temperature for

a given sample of gas, we can calculate the number of moles present:

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